Stability demands that the poles be situated in the lower half plane. This implies that \(\Delta _{\chi } < c/6\) . This is consistent for our purpose since the motivation for studying the finite temperature corrections on the boundary lies in studying the effect of light matter fields on the bulk. This also satisfies Hellerman's bound on the lightest non-vacuum primary [1]}. If \(\Delta _{\chi }\)  touches \(c/6\)  then we enter non-perturbative regime, and we cannot use anymore the HLLH monodromy answer.
